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Mathematical Definition

\begin{large}\[ \vec{v} = \frac{d\vec{r}}{dt}\]\end{large}

Representing Velocity Graphically

Velocity is commonly represented graphically in several ways:

• On a velocity versus time graph.
• As the spacing between points in a motion diagram.
• As the slope of a position versus time graph.

Average Velocity

Mathematical Definition

In mechanics, the term "average velocity" will almost always be used to denote the time-averaged velocity. The general defnition of the time average of a function

\begin{large}\[ f(t)\]\end{large}

is:

\begin{large}\[ \langle f(t)\rangle_{t} \equiv \frac{\int_{t_{i}}^{t_{f}} f(t)\:dt}{t_{f} - t_{i}} \]\end{large}

.

In the special case of velocity, this expression becomes:

\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\int_{t_{i}}^{t_{f}} \vec{v} \:dt}{t_{f}-t_{i}}
=\frac{\int_{t_{i}}^{t_{f}} \left(\frac{dx}{dt}\hat{x} + \frac{dy}{dt}\hat{y} + \frac{dz}{dt}\hat{z}\right) \:dt}{t_{f}-t_{i}}
\]\end{large}

We can now formally split the numerator into three integrals and make a change of variables in each of the integrals. Noting that (by the chain rule):

\begin{large}\[ dx = \frac{dx}{dt}\:dt\]\end{large}

with the corresponding expressions for dy and dz, we have:

\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\int_{x_{i}}^{x_{f}} \hat{x}\:dx + \int_{y_{i}}^{y_{f}} \hat{y}\:dy
+ \int_{z_{i}}^{z_{f}} \hat{z}\:dz}{t_{f} - t_{i}} \]\end{large}

These integrals are extremely simple, and lead to the very simple final expression:

\begin{large}\[ \langle \vec{v} \rangle_{t} = \frac{\vec{r}_{f} - \vec{r}_{i}}{t_{f}-t_{i}} \equiv \frac{\Delta\vec{r}}{\Delta t} \]\end{large}

Thus, the average velocity is simply the total change in position for a trip divided by the total elapsed time for the trip.

A One-Dimensional Example

Consider an example of one-dimensional motion. Suppose a student rushes from their dorm to the physics building in 2 minutes. After spending 4 minutes turning in their homework, the student hurries to the cafeteria in 2 minutes. The student eats lunch for 12 minutes, then walks to the library in 6 minutes. During which portion of the trip was the student moving the fastest?

For simplicity, imagine a school where all these buildings are on the same street. The street runs east to west. Suppose that the physics building is two blocks east of the dorm, the cafeteria is one block west of the dorm, and the library is three blocks east of the dorm. Before performing any calculations to characterize this trip, it is necessary to set up a coordinate system. One possibility, which we will use in this example, is shown below.

We are now ready to return to our question: in which portion of the trip was the student moving the fastest? The average velocity for a trip in one dimension will be defined as:

\begin{large} \[ \langle v\rangle_{t} = \frac{x_{\rm f} - x_{\rm i}}{t_{\rm f} - t_{\rm i}} \] \end{large}

To see how this equation works, consider the first part of the student's trip. In that part, the student moved from the dorm to the physics building in a time of 2 minutes. To evaluate the average velocity for this part, we simply substitute into the equation:

\begin{large} \[ \langle v_{\rm pd}\rangle_{t} = \frac{x_{\rm p} - x_{\rm d}}{t_{\rm p} - t_{\rm d}} = \frac{ +2\:{\rm blocks} - 0\:{\rm blocks}}{2\:{\rm minutes}} = + 1\:{\rm blocks/min}\]\end{large}

where we have used the subscript "p" to stand for the physics building and "d" for the dorm.

\begin{large}\[ \langle v \rangle_{t} = \frac{x_{\rm f} - x_{\rm i}}{t}\]\end{large}

We can compare this to the average velocity for the second trip made (from the physics building to the cafeteria):

\begin{large}\[ \langle v_{\rm cp}\rangle_{t} = \frac{x_{\rm c} - x_{\rm p}}{t_{\rm cp}} = \frac{(-1\:{\rm blocks})- (+ 2\:{\rm blocks})}{2\:{\rm minutes}} = -1.5\:{\rm blocks/min}\] \end{large}

The first thing to note here is that our answer has come out with a negative sign. For the first leg of the trip, the student has a velocity of + 1 block/min, and for the second leg, a velocity of - 1.5 blocks/min. These signs indicate the direction of the students motion, just as the sign of the position difference did. When reporting average velocities, it is a good practice to explicitly give the meaning of the signs, so that people do not have to be familiar with your specific coordinate system to understand the result. Thus, in this case, a more general way to report the student's movement is to say that for the first leg the average velocity was 1 block/min east, and for the second leg the average velocity was 1.5 blocks/min west. When the direction is included, the sign is removed.